## Introduction

summclust handles cluster robust variance estimation of linear regression models of the form

\begin{align} y &= X \beta + u \\ &= \begin{bmatrix} y_{1} \\ y_{2} \\ ...\\ y_{G} \end{bmatrix} = \begin{bmatrix} X_{1} \\ X_{2} \\ ...\\ X_{G} \end{bmatrix} \beta + \begin{bmatrix} u_{1} \\ u_{2} \\ ...\\ u_{G} \end{bmatrix}, \end{align}

with $$E(u|X) = 0$$, where group $$g$$ contains $$N_{g}$$ observations so that $$N = \sum_{g = 1}^{G} N_{g}$$. The regression residuals $$u$$ are allowed to be correlated within clusters, but are assumed to be uncorrelated across clusters. In consequence, the models’ covariance matrix is block diagonal. For each cluster, we denote $$E(u_{g} u_{g}') =\Sigma_{g}$$. We denote the predicted residual as $$\hat{u} = y - X \beta$$.

## Three Cluster-Robust Variance Estimators

The literature on cluster robust inference has proposed three different estimators, which all follow the same ‘sandwich’ structure

\begin{align} \hat{V}(\hat{\beta}) &= m (X'X)^{-1} (\sum_{g=1}^{G} \hat{\Sigma}_{g} ) (X'X)^{-1} \\ &= m (X'X)^{-1} (\sum_{g=1}^{G} \hat{s}_{g} \hat{s}_{g}') (X'X)^{-1} \end{align}

The three proposed types of CRV estimators vary in how $$\hat{s}_{g}$$ is estimated and in how small sample corrections $$m$$ are applied.

The most common cluster robust estimator, the CRV1 estimator, is defined as

\begin{align} \hat{s}_{g} = X_{g}'\hat{u}_{g} && \& && m = \frac{N-1}{N-k} / \frac{G}{G-1} \end{align}

When each cluster contains a unique observation, i.e. when $$G=N$$, the CRV1 estimator is equivalent to the HC1 heteroskedasticity robust estimator.

The CRV2 estimator is computed as

\begin{align} \hat{s}_{g} = X_{g}' M_{gg}^{-1/2} \hat{u}_{g} && \& && m = 1. \end{align}

To define $$M_{gg}$$, we first define the “hat” matrix

$$$H_g = X_g (X'X)^{-1} X_g'$$$

and $$I_{N_g}$$ as the diagonal matrix for all observations in cluster $$g$$.

$$M_{gg}$$ is then defined as

$$$M_{gg} = I_{N_g} - H_g.$$$

Last, the CRV3 estimator is defined as

\begin{align} \hat{s}_{g} = X_{g}' M_{gg}^{-1} \hat{u}_{g} && \& && m = G/(G-1). \end{align}

## Jackknife formulation of the CRV3 estimator (CRV3J)

Building on work by Niccodemi and Wansbeek, MacKinnon, Nielsen and Webb show that the CRV3 estimator can be computed as a Jackknife estimator.

First, let’s define $$\hat{\beta}^{(g)}$$, the OLS estimate of (1) when cluster g is omitted:

$$$\hat{\beta}^{(g)} = ((X'X)^{-1} - (X_{g}'X_{g})^{-1})(X'y - X_{g}'y_{g}), g = 1, ... , G.$$$

MNW show the the CRV3 estimator is equivalent to computing

$$$\hat{V}_{3}(\hat{\beta}) = \frac{G}{G-1} \sum{g = 1}^{G} (\hat{\beta}^{(g)} - \hat{\beta}) (\hat{\beta}^{(g)} - \hat{\beta})'$$$

They further propose the following Jackknive estimator, CRVJ:

$$$\hat{V}_{3J}(\hat{\beta}) = \frac{G}{G-1} \sum{g = 1}^{G} (\hat{\beta}^{(g)} - \bar{\beta}) (\hat{\beta}^{(g)} - \bar{\beta})'$$$

with $$\bar{\beta} = G^{-1} \sum_{g=1}^{G} \hat{\beta}^{(g)}$$.

When $$G=N$$, this Jackknife estimator is equivalent to the HC3 heteroskedasticity-robust estimator as proposed in MacKinnon and White.